Mathematics > Integral Calculus -7

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Detailed Solution for Integral Calculus -7 - Question 1

Integral Calculus -7 - Question 2

The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1 - cos t) between (0, 0) and (π, 2) is:

Detailed Solution for Integral Calculus -7 - Question 2

The given curve is

x(t) = (t + sin t), y(t) = (1 - cos t)

Integral Calculus -7 - Question 3

Which of the following is/are true?

1.

2.

3.

4. If f(x) ≥ φ(x) and both functions are integrable in [a, b]

Integral Calculus -7 - Question 4

is revolved about major and minor axis respectively. Then the ratio of these solids made by two revolutions is:

Detailed Solution for Integral Calculus -7 - Question 4

Case (i): When ellipse is rotated about major axis:

Take a small disc at a length of x from the centre of thickness dx. Then the volume of solid obtained by rotation will be ∫(−a to a)(Area)dx

Area of disc =πr^{2}

r can be calculated from the equation of ellipse as

x^{2}/a^{2} + r^{2}/b^{2 }= 1

⇒ r^{2 }= b^{2}(1−x^{2}/a^{2}) = b^{2}/a^{2}(a^{2}−x^{2})

∴Volume of major axis = ∫(−a to a)πa^{2}/b^{2}(a^{2}−x^{2})dx

= [πb^{2}x − πb^{2}x^{3}/3a^{2}](−a to a)

= 4/3πab^{2}

Case (ii): When ellipse is rotated about minor axis:

Following similar procedure as case (i),

r^{2 }= a^{2}/b^{2}(b^{2 }− y^{2})

In this case, the area will be integrated w.r.t dy as it is rotated about the Y-axis.

∴Volume of minor axis = ∫(−a to a)πa^{2}/b^{2}(b^{2}−y^{2})dx

= [πa^{2}y − πa^{2}y^{3}/3b^{2}](−a to a)

= 4/3πab^{2}

∴ Volume about major axis/Volume about minor axis

= b/a

Detailed Solution for Integral Calculus -7 - Question 5

...(i)

...(ii)

Integral Calculus -7 - Question 6

The surface area of the segment of a sphere of radius a and height h is given by:

Detailed Solution for Integral Calculus -7 - Question 6

Let the sphere be generated by the revolution about the x - axis of the circle

x^{2} + y^{2}=a^{2} ...(i)

Let OA =a, OC = b and OB = b + h

Hence The required surface

Detailed Solution for Integral Calculus -7 - Question 7

Integral Calculus -7 - Question 8

The moment of inertia about the axis of y of the region in the xy-plane bounded by y = 4 - x^{2} and the x-axis, is, proportional to

Detailed Solution for Integral Calculus -7 - Question 8

The given curve is y = 4x^{2}

Hence Required moment of inertia

Detailed Solution for Integral Calculus -7 - Question 9

...(i)

...(ii)

Detailed Solution for Integral Calculus -7 - Question 10

Detailed Solution for Integral Calculus -7 - Question 11

Given that

y = log_{e}

Integral Calculus -7 - Question 12

The intrinsic equations of the cardioids r = a (1 - cos θ) and r = a(1 + cos θ) measured from the pole are:

Detailed Solution for Integral Calculus -7 - Question 12

The equations of given cardioids are

r= a (l -cos θ) ...(i)

r= a (1 + cos θ) ...(ii)

Intrinsic equation for cardioid (i):

...(iii)

Integral Calculus -7 - Question 13

Consider the Assertion (A) and Reason (R) given below:

Assertion (A)

Reason (R) - sin x is continuous in any closed interval [ 0 ,t] .

The correct answer is:

Detailed Solution for Integral Calculus -7 - Question 14

Detailed Solution for Integral Calculus -7 - Question 15

Let

Integral Calculus -7 - Question 16

The perimeters of the cardioids r = a (1 - cos θ) and r = a (1 + cos θ) differ by:

Detailed Solution for Integral Calculus -7 - Question 16

The curves are

r = a(1 - cos θ) ...(i)

r = a(1 + cos θ) ...(ii)

The perimeter for curve (i) is

Hence s_{1} - s_{2} = 8a - 8a = 0.

Integral Calculus -7 - Question 17

The length of the arc of the curve 6xy = x^{4} + 3 from x = 1 to x = 2 is:

Detailed Solution for Integral Calculus -7 - Question 17

The given curve is

6xy = x^{4} + 3

Thus Length of the arc between x

= 1 and x = 2 is:

Integral Calculus -7 - Question 18

The area of the region in the first quadrant bounded by the y-axis and curves y = sin x and y = cos x is:

Detailed Solution for Integral Calculus -7 - Question 18

The given curves are

y = sin x ...(i)

y = cos x ...(ii)

The curves intersect at a point B

Hence Required area = Area OABO

Integral Calculus -7 - Question 19

The line y = x + 1 is revolved about x-axis. The volume of solid of revolution formed by revolving the area covered by the given curve, x-axis and lines x = 0, x = 2 is:

Detailed Solution for Integral Calculus -7 - Question 19

The given curve is y = x + 1. This represents a straight line

Hence The required volume

Integral Calculus -7 - Question 20

The length of the complete cycloid x = a (θ + sin θ), y = a(1 - cos θ) is given by:

Detailed Solution for Integral Calculus -7 - Question 20

The equations of the curve are

x = a(θ + sin θ), y = a(1 - cos θ)

Hence The entire length of the curve

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